Inequalities

Just as we can make statements using a language such as English, we may also make statements using mathematics. For instance, if we say that “doubling Tina’s age will give us 60,” we could also write this mathematically with symbols:

Here the variable t stands in for Tina’s unknown age. This is what we call an equation and it is merely a statement made by separating two expressions with an equals sign. We can solve for Tina’s age by dividing both sides of the equation by 2 to get t = 30.

Sometimes, however, we want to claim that there is a relationship between two quantities, but that the quantities are not necessarily equal. For this we need to use inequalities rather than equations. For instance, if we say that “Nathan swam less than 2500 yards,” the same statement could be written mathematically in symbols as:

Here the variable n stands for the number of yards that Nathan swam. Additionally, if one were to make the slightly modified statement that Nathan swam no more than 2500 yards, the mathematical statement would read

Note that the single bar under the less than sign indicates that Nathan could have also swum exactly 2500 yards.

Similarly, the statement “Jana’s dog weighs more than 20 pounds,” could be expressed

where d represents the weight of Jana’s dog.

Just as we can solve equations (in the first example above when we divided by 2 to get Tina’s age), we can also solve inequalities. Let’s look at an example.

Example 1

The statement, “$1,000 less than twice Will’s current income would be greater than $35,000,” could be written in this way:

where w represents Will’s current income.

How big, comparatively speaking, is Will’s current income?

Solution 1

An inequality like this can be solved in almost exactly the same way as a linear equation (see the tutorial entitled Solving Linear Equations for a review of this topic) with a few caveats that we will address later. We treat the inequality sign as though it were an equals sign and first work on collecting all the constant terms to one side and all the terms which include w on the other. To do this we just add 1000 to both sides and get

Then we just divide both sides by 2 yielding,

While at this point, we may not know exactly what Will’s current income is, we can conclude that it is something greater than $18,000.

There is one potential problem with this method. We assumed that the rules of algebra are the same for equations as they are inequalities. Here’s a simple example to show that dealing with inequalities can be slightly more complicated.

Consider the following claim:

If we were to naively divide both sides by -2 and nothing more, we would arrive at the statement that . But, let’s try one particular number greater than -3 in the first inequality, say 4. Then we have the statement, or simplified, which is clearly false.

Let’s think about why dividing or multiplying both sides of an inequality is problematic. Suppose we have –x < 2. This means that we are looking for all values of x such that the inequality holds true. Certainly any positive x-value would satisfy the inequality, because it’s opposite would be less than 0, which is certainly less than 2. In fact, any x-value that is greater than -2 would satisfy the inequality –x < 2. In other words, the solution to the inequality is x > -2, not x < -2! Try several to convince yourself! So what, do we need to do to accommodate situations such as -2x > 6?

When we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes. So in the above example, we should get the solution . It is easy to check that for numbers less than -3, the original inequality holds.

Another thing to keep in mind is that caution must be used when solving inequalities that contain terms raised to powers. For instance, if we have the inequality , taking the square root of each side does not result in x > 4. In fact, it results in | x | > 4 because = | x |. Think about it: if x2 = 9, for example, then x could be 3 or -3. The solution to | x | > 4 has to be x > 4 or x < -4. These are exactly the numbers that satisfy the inequality .

Example 2

Solve this inequality:

Solution2

First, we distribute the 3 and get,

Then, we collect all the constant terms on the right hand side by adding 12 to both sides and collect all the x terms on the left hand side by subtracting an x from each side:

Simplifying yields,

And then dividing both sides by 2 gives,

Note that the sign did not change direction since we divided by positive 2.